448_04NetPV - Lecture Notes 4 Net Present Value The single...

28Lecture Notes 4Net Present ValueThe single period case• Recall from the previous set of notes that, when the interest rate per period is r, the net present value of an investment that has a negative cash flow of –C0now and a positive cash flow one period in the future of C1is.(1)• The amount C1/(1+r) is the present valueof the cash flow that will be earned next period.• Conversely, we could calculate everything in terms of next period’s values. The future valueof the cash investment this period will be –C0(1+r).Multiple periods• When the interest earned on an investment is re-invested, the interest compounds. An investment of C0becomes C0(1+r) after the first year. If this entire sum is re-invested, the value at the end of two years will be C0(1+r)2. More, generally the future value of the investment after Tyears of earning compound interestwill be(2)• Conversely, the present value of a cash flow Cthat is to be received Tperiods into the future is(3)The PVrepresents the amount of money that would have to be invested now at the compound interest rate of rper period in order to obtain, in Tperiods, the same cash flow of C. The term multiplying Cin (3),NPVC0–C11r+---------+=FVC01r+()T=PVC1r+T----------------=

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29(4)is known as the present value factor.• A streamof cash flows (–C0, C1, C2, …, CT), representing an investment opportunity that will cost money immediately but will yield positive amounts in each of Tfuture periods, will have a net present value(5)• Often one sees investments where there is a stated annual interest ratebut compounding occurs more frequently than once a year. For example, most mortgages have interest paid at monthly in-tervals but the return is given as a stated annual interest rate. The future value of an investment C0with a stated annual interest rate of rbut compounding for mperiods within the year will be(6)The effective annual interest rate

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